The 𝑑₂-transformation for infinite double series and the 𝐷₂-transformation for infinite double integrals

Greif, Chen; Levin, David

doi:10.1090/S0025-5718-98-00955-7

The $d_2$-transformation for infinite double series and the $D_2$-transformation for infinite double integrals
HTML articles powered by AMS MathViewer

by Chen Greif and David Levin PDF

Math. Comp. 67 (1998), 695-714 Request permission

Abstract:

New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate $d$- and $D$-transformations. The $D_2$-transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the $d_2$-transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given.

References

J. S. R. Chisholm, Rational approximants defined from double power series, Math. Comp. 27 (1973), 841–848. MR 382928, DOI 10.1090/S0025-5718-1973-0382928-6
Annie Cuyt, Padé approximants for operators: theory and applications, Lecture Notes in Mathematics, vol. 1065, Springer-Verlag, Berlin, 1984. MR 750977, DOI 10.1007/BFb0099706
Annie Cuyt, Multivariate Padé approximants revisited, BIT 26 (1986), no. 1, 71–79. MR 833832, DOI 10.1007/BF01939363
A. Cuyt, Extension of: “A multivariate convergence theorem of the ‘de Montessus de Ballore’ type” to multipoles [J. Comput. Appl. Math. 32 (1990), no. 1-2, 47–57; MR1091774 (92b:65015)], J. Comput. Appl. Math. 41 (1992), no. 3, 323–330. MR 1187668, DOI 10.1016/0377-0427(92)90139-O
C. Greif, 1994 Singularity detection and bivariate generalization of the $d$ and $D$ transformations, M.Sc. Thesis, Tel Aviv Univ.
D. Levin, 1971 Development of non-linear transformations of series and sequences to increase rate and domain of convergence, and their use for computing results from formal solutions to applied math. problems, M.Sc. Thesis, Tel Aviv Univ.
David Levin, Development of non-linear transformations of improving convergence of sequences, Internat. J. Comput. Math. 3 (1973), 371–388. MR 359261, DOI 10.1080/00207167308803075
—, 1975 Methods for accelerating convergence of infinite series and integrals, Ph.D. Thesis, Tel Aviv Univ.
D. Levin, General order Padé-type rational approximants defined from double power series, J. Inst. Math. Appl. 18 (1976), no. 1, 1–8. MR 433087, DOI 10.1093/imamat/18.1.1
David Levin, On accelerating the convergence of infinite double series and integrals, Math. Comp. 35 (1980), no. 152, 1331–1345. MR 583511, DOI 10.1090/S0025-5718-1980-0583511-3
David Levin and Avram Sidi, Two new classes of nonlinear transformations for accelerating the convergence of infinite integrals and series, Appl. Math. Comput. 9 (1981), no. 3, 175–215. MR 650681, DOI 10.1016/0096-3003(81)90028-X
Avram Sidi and David Levin, Rational approximations from the $d$-transformation, IMA J. Numer. Anal. 2 (1982), no. 2, 153–167. MR 668590, DOI 10.1093/imanum/2.2.153
A. Sidi, 1977 Exponential function approximation to Laplace transform inversion and development of non-linear methods for accelerating the convergence of infinite integrals and series, Ph.D. Thesis, Tel Aviv Univ.
Avram Sidi, Convergence properties of some nonlinear sequence transformations, Math. Comp. 33 (1979), no. 145, 315–326. MR 514827, DOI 10.1090/S0025-5718-1979-0514827-6
A. Sidi, Some properties of a generalization of the Richardson extrapolation process, J. Inst. Math. Appl. 24 (1979), no. 3, 327–346. MR 550478, DOI 10.1093/imamat/24.3.327
Avram Sidi, Analysis of convergence of the $T$-transformation for power series, Math. Comp. 35 (1980), no. 151, 833–850. MR 572860, DOI 10.1090/S0025-5718-1980-0572860-0
Avram Sidi, Convergence analysis for a generalized Richardson extrapolation process with an application to the $d^{(1)}$-transformation on convergent and divergent logarithmic sequences, Math. Comp. 64 (1995), no. 212, 1627–1657. MR 1312099, DOI 10.1090/S0025-5718-1995-1312099-5
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1